Choose the native relativity of simultaneity preset. It contains two flashes at rest frame time coordinate five, at spatial coordinates minus four and plus four. Set the velocity slider first to zero, then to zero point five times the speed of light. The Lorentz formula subtracts beta times position from time, then multiplies by gamma. At zero velocity both transformed times remain five. At beta one half, gamma is approximately one point one five four seven. The left flash has transformed time approximately eight point zero eight two nine, while the right flash has time three point four six four one. The exact held plotted squares use twenty four pixels per coordinate unit in both panels. Blue is the left rest frame flash and pink is the right flash. Their unequal boosted times follow the position dependent term, not a different light signal speed. These are coordinates in the source units, with the speed of light set to one. Load the native causality preset and inspect its Elsewhere event, time coordinate two and position seven. Its interval relative to the origin is four minus forty nine, or minus forty five. Compare native velocity zero with zero point nine. Initially its time coordinate is positive two. At beta zero point nine, the transformed time becomes negative, approximately minus nine point eight six four five, while the transformed position is approximately eleven point nine three zero. Both panels use fifteen pixels per coordinate unit, with transformed time increasing upward. The event moves below the time zero axis in boosted coordinates. Independently squaring the new coordinates still gives interval minus forty five. This is a spacelike separation from the origin, so changing its coordinate time order does not reverse an allowed causal signal. The page interval classifier concerns each event relative to the origin; it is not automatically the interval between two arbitrary selected events. Select the native time dilation preset. Its first two ticks occur at time coordinates three and six, both at position zero in the rest frame S. That is a clock at rest in S, so its proper tick interval is three in the source units. Set native beta to zero point six, then zero point eight. Because the ticks have no rest frame position difference, their moving frame time interval is gamma times three. Gamma rises from one point two five to five thirds. The moving frame interval therefore grows from three point seven five to five, while the clock proper interval remains three. All bars use seventy pixels per interval unit. The proper interval divided by the moving frame interval falls from zero point eight to zero point six, exactly the displayed one over gamma formula. This explanation applies to this specified inertial clock worldline. It does not infer elapsed proper time from an arbitrary isolated event or claim a complete twin paradox calculation through acceleration.